IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 3, MARCH 2005 1093 Effective Medium Theory of Magnetization Reversal in Magnetically Interacting Particles Heliang Qu and JiangYu Li Abstract—We report an effective medium theory of magnetiza- One of the breakthroughs was made about ten years ago tion reversal and hysteresis in magnetically interacting particles, when an exchange coupling mechanism was proposed to en- where the intergranular magnetostatic interaction is accounted hance the energy product of permanent magnets [3]. The idea for by an effective medium approximation. We introduce two is to mix the magnetic hard and soft phases at nanoscale so dimensionless parameters, and H, that completely charac- terize the hysteresis in a ferromagnetic polycrystal when the that they can be coupled through the intergranular exchange grain size is much larger than the exchange length so that the interaction, leading to dramatically enhanced remanence and exchange coupling can be ignored. The competition between the energy product. This again highlights the importance of inter- anisotropy energy and the intergranular magnetostatic energy granular magnetic interactions, and calls for new models that is measured by , while the competition between the anisotropy can take those interactions into account. While micromagnetic energy and Zeeman’s energy is measured by H. The hysteresis loop, magnetostatic energy density, and anisotropy energy density simulations are able to explain the remanence enhancement in calculated by using this theory agrees well with micromagnetic exchange-spring magnets [4], they are often computationally simulations. The calculations also reveal that the subnucleation extensive and time-consuming. There are also earlier efforts to field switching due to the magnetic field fluctuation is important modify the Stoner–Wohlfarth model by Atherton and Beattie when the magnet is not very hard, and that has been accounted using a mean field correction [5], where a magnetic field pro- for by a probability-based switching model. portional to the average magnetization is introduced on top of Index Terms—Effective medium approximation, hysteresis, in- the applied field to account for the magnetic interactions among tergranular magnetostatic interaction, magnetization reversal. grains, although they have not sought to determine the propor- tional parameter introduced for the correction in the modified I. INTRODUCTION model. We hope to address these issues here by developing an effective medium theory to analyze the magnetization reversal TONER and Wohlfarth’s celebrated model [1] has been in magnetically interacting particles. Our preliminary results S widely used for more than 50 years to analyze the mag- on remanence enhancement in polycrystalline magnets have netization reversal in polycrystalline permanent magnets. The been reported in [6], and in this paper magnetization reversal underlying assumption of the Stoner–Wohlfarth model is that and hysteresis loops will be analyzed. We will focus on the each grain in the polycrystal reverses its magnetization through magnetization reversal through coherent rotation only, and thus rotation without interacting with each other, and as a result, the will not consider motions and pinnings of domain walls [7]. remanent magnetization is simply given by the volume average of magnetization over the polycrystal II. ENERGETICS AND EFFECTIVE MEDIUM THEORY (1) We consider an assemblage of single-phase single-domain magnetic particles, with particle size smaller than the single where is saturation magnetization and is the angle between domain limit so that no domain wall movement is involved, the remanent magnetization of the polycrystal and the easy axis yet large enough compared to the exchange length so that the of the grain. While the model provides an accurate estimate of short-range exchange coupling can be ignored. As a result, the the remanence of most permanent magnets, it also highlights the potential energy of the assemblage is given by importance of intergranular magnetic interactions in the design and optimization of new magnetic materials. Permanent mag- (2) nets with large remanence and coercivity are desirable for en- ergy storage due to their high energy product [2], and in order to where the first term is the uniaxial anisotropy energy at the enhance the energy product of polycrystalline magnets, the in- lowest order, with being the angle between the local magnetic tergranular magnetic interactions must be strengthened, so that polarization and the easy axis of the particles; the second term higher remanence than Stoner–Wohlfarth’s prediction can be is Zeeman’s energy, where is the applied magnetic field; and accomplished. the third term is the magnetostatic energy, where is the de- magnetizing field induced by the distribution of in , the re- Manuscript received February 3, 2004; revised November 20, 2004. This gion occupied by the assemblage. Here, is a function of work was supported by the Nebraska Research Initiative. which we seek to characterize at different , while is used The authors are with the Department of Engineering Mechanics, University of to label different grains at different orientations. The configura- Nebraska-Lincoln, Lincoln, NE 68588-0526 USA (e-mail: [email protected]; [email protected]). tion in a typical grain at orientation is schematically shown in Digital Object Identifier 10.1109/TMAG.2004.843318 Fig. 1. 0018-9464/$20.00 © 2005 IEEE 1094 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 3, MARCH 2005 for given distribution of using Green’s function method [8] (7) where is the magnetic Green’s function, and denotes the gradient with respect to . To carry out the analysis further, we recall that the grain size of the polycrystal is smaller than the single domain limit, and thus the magnetic polarization in indi- vidual grain is uniform. In addition, we assume that grains with identical orientation will have identical magnetic polarization, an effective medium approximation, so that (8) where is the number of different orientations and (9) Fig. 1. The configuration for a grain at orientation . otherwise is the characteristic function of a region containing all grains We consider the long-range magnetostatic energy first. In mi- having orientation , in which magnetic polarization is uni- cromagnetic simulation, calculating the magnetostatic energy form. Multiplying (7) by leads to often takes most of the CPU time due to its nonlocality. We in- tend to address this problem using an effective medium approx- imation which is believed to be accurate and computationally (10) efficient. To this end, we decompose the magnetic polarization and magnetostatic field into two parts, and rewrite the magneto- static energy as where is the two-point correlation function of grain distribution in the polycrystal, which gives the probability of locating grain orientation at and grain orien- (3) tation at simultaneously. When the two-point correlation function is ellipsoidal [9], we have where the overhead bar is used to represent the volume averaged quantities, and the primed ones are variations from the averages. Clearly, the cross products vanish because the volume integra- tions of primed field variables are zero. Notice that we have (11) (4) with where is the magnitude of the macroscopic magnetic po- (12) larization along the applied field direction, is the saturation magnetic polarization, and is a dimensionless parameter intro- duced to represent the macroscopic magnetization in the poly- crystal. In addition where while is the dimension of principal axis of the ellipsoid, and is the volume fraction of grain or the probability of locating grain at . In fact, is (5) a second rank tensor that can be regarded as the demagnetizing factor of grains depending on their shape. Integrating (11) over is the magnetic field induced by a uniform magnetic polarization the polycrystal , we obtain in , where is the demagnetizing factor depending on the shape of . For in-plane magnetization in an ellipsoidal with (13) dimensions , is small. The magnetic field induced by the magnetostatic interaction which is the intergranular magnetostatic field we try to charac- among particles, , can be determined by solving Maxwell’s terize. For spherical distribution, , which leads to equation (6) (14) QU AND LI: THEORY OF MAGNETIZATION REVERSAL IN MAGNETICALLY INTERACTING PARTICLES 1095 where is the volume of , and is the in-plane demagne- are constrained within plane for simplicity in the fol- tizing factor. Even though is small for thin-plate magnets, lowing analysis, although the theory can be applied to three-di- it cannot be ignored for nonhard magnetic materials. The treat- mensional texture without difficulty. With such assumption, the ment for more general grain shape is straightforward. Notice orientational averaging in the polycrystal can be carried out as that the first term of the magnetostatic energy is a constant and thus is irrelevant in the following energy minimization and will (22) be ignored. It is emphasized that we have derived the interac- tion field analytically using the effective medium approxima- tion, while in Atherton and Bettie [5] this interaction field is and introduced phenomenologically. The total anisotropy energy of the system can be determined (23) as (15) where is the orientation distribution function (ODF) that and Zeeman’s energy can be evaluated as describes the probability of locating a grain at a particular ori- entation, which can be an arbitrary function of [11],